Upper and lower limits

The upper (lower) limit of a sequence is the largest (respectively, smallest) limit among all partial (finite and infinite) limits of a given sequence of real numbers. For any sequence of real numbers , the set of all its partial limits (finite and infinite) on the extended number axis (i.e. in the set of real numbers, completed by the symbols and ) is non-empty and has both a largest and a smallest element (finite or infinite). The largest element of the set of partial limits is said to be the upper limit (lim sup) of the sequence and is denoted by

while the smallest element is said to be the lower limit (lim inf) and is denoted by

For instance, if

then

If

then

If

then

Any sequence has a lim sup and a lim inf, and if the sequence is bounded from above (from below) its lim sup (lim inf) is finite. A number is the lim sup (lim inf) of a sequence , if and only if for any the following conditions are fulfilled: a) there exists a number such that for all indices the inequality () is true; b) for any index there exists an index such that and (). The meaning of condition a) is that for a given there exists in the sequence only a finite number of terms such that (). The meaning of condition b) is that there exists an infinite set of terms such that (). Provided both are finite, the lim inf can be reduced to the lim sup by changing the signs of the terms of the sequence:

For a sequence , to have a limit (finite or infinite (equal to one of the symbols or )) it is necessary and sufficient that

The upper (lower) limit of a function at a point is the limit of the upper (lower) bounds of the sets of values of in a neighbourhood of , when this neighbourhood contracts towards . These limits are denoted by

Let the function be defined on a metric space and assume real values. If and is an -neighbourhood of , , then

and

At each point the function has both an upper limit and a lower limit (finite or infinite). The function is upper semi-continuous, while the function is lower semi-continuous on (in the sense of the concept of semi-continuity of functions which assume values in the extended number axis, cf. also Semi-continuous function).

For a function to have a finite or infinite (equal to or ) limit at a point it is necessary and sufficient that

The concept of the upper limit (lower limit) of a function at a point can be naturally extended to real-valued functions defined on topological spaces.

The upper and lower limit of a sequence of sets , are the set

consisting of the elements which belong to an infinite number of sets , and the set

of the elements which belong to all sets , starting from some index , respectively. Obviously, .

References

[1]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[2]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[3]	L.D. Kudryavtsev, "A course of mathematical analysis" , 1 , Moscow (1988) (In Russian)
[4]	S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)
[5]	F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))

Comments

The upper limit is also called the limes superior or limit superior, and the lower limit the limes inferior or limit inferior. Cf. also Upper and lower bounds.

The limit superior and limit inferior of a sequence of subsets of a set are given by the formulas

References

[a1]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)